Karoubi's relative Chern character, the rigid syntomic regulator, and the Bloch-Kato exponential map

TL;DR

This paper develops a new variant of Karoubi's relative Chern character for p-adic schemes, compares it with the rigid syntomic regulator, and relates it to the etale p-adic regulator via the Bloch-Kato exponential map, extending previous results.

Contribution

It introduces a novel relative Chern character for p-adic schemes and establishes its relation to existing regulators, generalizing prior work to all smooth projective schemes.

Findings

Constructed a variant of Karoubi's relative Chern character.

Proved comparison with the rigid syntomic regulator.

Related the Chern character to the etale p-adic regulator via Bloch-Kato exponential.

Abstract

We construct a variant of Karoubi's relative Chern character for smooth, separated schemes over the ring of integers in a p-adic field and prove a comparison with the rigid syntomic regulator. For smooth projective schemes we further relate the relative Chern character to the etale p-adic regulator via the Bloch-Kato exponential map. This reproves a result of Huber and Kings for the spectrum of the ring of integers and generalizes it to all smooth projective schemes as above.